Whether or not the administation acted rashly is hardly in dispute. But it seems fair to say she has a good chance of proving he libeled her. There was obvious malicious intent from the way the video was edited, and she suffered legitmate harm (in a legal sense) from his claim. He'd be wise to argue she's a public figure and thus the burden of proof is higher, but how dumb the administation was is sort of irelevant. She was still fired (or forced to resign) because of a lie he told with intent to harm, and from the way the video was selectively edited, it doesn't seem to hard to prove he knew it was a lie. That is libel.